ریاضیات

Formal Examples of Charts

1. The Real Projective Line (RP^1)

The projective line is defined as the set of lines through the origin in R^2. It cannot be covered by a single chart. We define two charts (U1, φ1) and (U2, φ2):

The transition map on the overlap is given by f(t) = 1/t. Since 1/t is C^∞ for t != 0, RP^1 inherits a smooth differentiable structure.

2. Graph of a Smooth Function

Let f: R^n → R be a smooth function. The graph G = {(x, f(x)) | x ∈ R^n} is an n-dimensional submanifold of R^(n+1).

This is known as the canonical chart for a graph. It demonstrates that any smooth function's graph is intrinsically a manifold.